For What Value Of X X X Is \cos (x) = \sin \left(14^{\circ}\right ], Where 0 ∘ \textless X \textless 90 ∘ 0^{\circ} \ \textless \ X \ \textless \ 90^{\circ} 0 ∘ \textless X \textless 9 0 ∘ ?A. 28 ∘ 28^{\circ} 2 8 ∘ B. 31 ∘ 31^{\circ} 3 1 ∘ C. 76 ∘ 76^{\circ} 7 6 ∘ D.

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Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we will focus on solving a specific type of trigonometric equation, where we need to find the value of $x$ that satisfies the equation $\cos (x) = \sin \left(14^{\circ}\right)$, given that $0^{\circ} \ \textless \ x \ \textless \ 90^{\circ}$. This equation involves the cosine and sine functions, which are two of the most important trigonometric functions.

Understanding Trigonometric Functions

Before we dive into solving the equation, let's briefly review the trigonometric functions involved. The cosine function, denoted by $\cos (x)$, is a periodic function that represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The sine function, denoted by $\sin (x)$, is also a periodic function that represents the ratio of the opposite side to the hypotenuse in a right-angled triangle.

Properties of Trigonometric Functions

To solve the equation, we need to use the properties of trigonometric functions. One of the most important properties is the co-function identity, which states that $\cos (x) = \sin (90^{\circ} - x)$. This identity allows us to rewrite the equation in terms of the sine function.

Rewriting the Equation

Using the co-function identity, we can rewrite the equation as follows:

$$\cos (x) = \sin \left(14^{\circ}\right)$$

$$\sin (90^{\circ} - x) = \sin \left(14^{\circ}\right)$$

Solving for $x$

Now that we have rewritten the equation, we can solve for $x$. Since the sine function is periodic, we can add or subtract multiples of $360^{\circ}$ to the angle without changing the value of the sine function. Therefore, we can rewrite the equation as follows:

$$\sin (90^{\circ} - x) = \sin \left(14^{\circ} + 360^{\circ}n\right)$$

where $n$ is an integer.

Finding the Value of $x$

To find the value of $x$, we need to equate the angles inside the sine functions. This gives us the following equation:

$$90^{\circ} - x = 14^{\circ} + 360^{\circ}n$$

Solving for $x$, we get:

$$x = 90^{\circ} - 14^{\circ} - 360^{\circ}n$$

Restricting the Value of $x$

Since we are given that $0^{\circ} \ \textless \ x \ \textless \ 90^{\circ}$, we need to restrict the value of $x$ to this range. This means that we need to find the value of $n$ that satisfies the inequality:

$$0^{\circ} \ \textless \ 90^{\circ} - 14^{\circ} - 360^{\circ}n \ \textless \ 90^{\circ}$$

Simplifying the inequality, we get:

$$-14^{\circ} - 360^{\circ}n \ \textgreater \ 0^{\circ}$$

$$-14^{\circ} - 360^{\circ}n \ \textless \ 90^{\circ}$$

Finding the Value of $n$

To find the value of $n$, we need to solve the inequality. Since $-14^{\circ} - 360^{\circ}n \ \textgreater \ 0^{\circ}$, we can divide both sides by $-360^{\circ}$ to get:

$$n \ \textless \ -\frac{14^{\circ}}{360^{\circ}}$$

Simplifying the inequality, we get:

$$n \ \textless \ -0.0389$$

Since $n$ is an integer, we can round down to the nearest integer to get:

$$n \ \textless \ -1$$

Finding the Value of $x$

Now that we have found the value of $n$, we can substitute it back into the equation for $x$:

$$x = 90^{\circ} - 14^{\circ} - 360^{\circ}n$$

Substituting $n = -1$, we get:

$$x = 90^{\circ} - 14^{\circ} - 360^{\circ}(-1)$$

Simplifying the equation, we get:

$$x = 90^{\circ} - 14^{\circ} + 360^{\circ}$$

$$x = 76^{\circ}$$

Conclusion

In this article, we have solved a trigonometric equation involving the cosine and sine functions. We have used the co-function identity to rewrite the equation in terms of the sine function, and then solved for $x$ using the properties of the sine function. We have also restricted the value of $x$ to the range $0^{\circ} \ \textless \ x \ \textless \ 90^{\circ}$, and found the value of $x$ to be $76^{\circ}$.

Final Answer

The final answer is $\boxed{76^{\circ}}$.

Introduction

In our previous article, we solved a trigonometric equation involving the cosine and sine functions. We used the co-function identity to rewrite the equation in terms of the sine function, and then solved for $x$ using the properties of the sine function. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q: What is the co-function identity?

A: The co-function identity is a property of trigonometric functions that states that $\cos (x) = \sin (90^{\circ} - x)$. This identity allows us to rewrite the equation in terms of the sine function.

Q: How do I use the co-function identity to rewrite the equation?

A: To use the co-function identity, simply replace the cosine function with the sine function and subtract $x$ from $90^{\circ}$. For example, if we have the equation $\cos (x) = \sin \left(14^{\circ}\right)$, we can rewrite it as $\sin (90^{\circ} - x) = \sin \left(14^{\circ}\right)$.

Q: What is the significance of the range $0^{\circ} \ \textless \ x \ \textless \ 90^{\circ}$?

A: The range $0^{\circ} \ \textless \ x \ \textless \ 90^{\circ}$ is important because it restricts the value of $x$ to the first quadrant of the unit circle. This means that the value of $x$ must be between $0^{\circ}$ and $90^{\circ}$.

Q: How do I find the value of $n$?

A: To find the value of $n$, we need to solve the inequality $n \ \textless \ -\frac{14^{\circ}}{360^{\circ}}$. Since $n$ is an integer, we can round down to the nearest integer to get $n \ \textless \ -1$.

Q: What is the final answer?

A: The final answer is $\boxed{76^{\circ}}$.

Frequently Asked Questions

Q: What is the difference between the sine and cosine functions?

A: The sine and cosine functions are two of the most important trigonometric functions. The sine function represents the ratio of the opposite side to the hypotenuse in a right-angled triangle, while the cosine function represents the ratio of the adjacent side to the hypotenuse.

Q: How do I use the properties of the sine function to solve the equation?

A: To use the properties of the sine function, simply apply the co-function identity and then solve for $x$ using the properties of the sine function.

Q: What is the significance of the unit circle?

A: The unit circle is a circle with a radius of 1 that is centered at the origin of the coordinate plane. It is used to represent the trigonometric functions and their properties.

Conclusion

In this article, we have provided a Q&A section to help clarify any doubts or questions that readers may have. We have also provided frequently asked questions to help readers understand the concepts and properties of trigonometric functions.

Final Answer

The final answer is $\boxed{76^{\circ}}$.